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Prime Numbers: Progress and Pitfalls

Speaking at the MAA Carriage House on November 11, Daniel Goldston (San Jose State University) offered insight into both prime numbers and the people who study them.

Goldston’s talk, "Prime Numbers: Progress and Pitfalls," explored the enduring appeal of the primes—and their knack for trickery.

Primes are popular with both mathematicians and the public, Goldston noted, but perhaps for different reasons. An amateur may investigate the primes because the numbers are familiar and easy to define.

"They’re not threatening," said Goldston. "They tend not to scare people away like sheaves of germs of C*-algebras or whatever."

A mathematician, on the other hand, studies the primes because he or she relishes a hard problem, according to Goldston. The apparently innocent indivisibles possess a seemingly boundless capacity to mystify.

"By studying primes you will achieve humility . . . Actually not, most of the mathematicians I know studying primes aren’t particularly good on humility," Goldston quipped. "But they get to experience lots of humiliation."

The primes have fooled Golston himself often enough, and even he can’t always gauge how difficult it will be to prove a given result.

Case in point: It is possible to find 1000 consecutive primes that end with the digits 876543211111. Though incredulous upon hearing this statement for the first time, Goldston might have known better. The impossible-sounding result follows from a 2000 paper by Daniel Shiu in the Journal of the London Mathematical Society.

"The funny thing is I had actually refereed the paper," Goldston told his Carriage House audience. The methods Shiu used were clever, Goldston remembered, but routine. "I didn’t realize it had this application."

Progress on prime numbers is slow, Goldston said, in part because even properties that appear obvious on inspection can resist proof for decades. Gauss conjectured the prime number theorem, which describes the distribution of prime numbers, in the 1790s, but the result wasn’t proven until 1896.

And even though mathematicians have been puzzling over them since at least 1977, everything that is known about what John Conway dubbed "jumping champions" can be presented on a single slide.

Given a number n, the jumping champion corresponding to n is the the most frequently occurring difference of consecutive primes less than or equal to n. The champions vary a bit for small n, but 6 emerges as a consistent winner past n = 947 and holds its solo title at least up to n = 1015. In 1999, A. A. Odlyzko, M. Rubinstein, and M. Wolf conjectured that 30 will take over as jumping champion at n = 1.7427*1035, and that 210 will start winning out at around 10425.

Goldston isn’t sure, though, whether the researchers themselves even still believe these estimates, which they obtained heuristically.

"Nothing has been proven," Goldston said.

Proofs about primes have recently made news, of course. In April 2013, relative unknown Yitang Zhang (University of New Hampshire) made stunning progress on the twin prime conjecture. Another of those statements mathematicians are pretty sure is true even if they can’t prove it, the twin prime conjecture asserts that there are infinitely many pairs of primes that differ by 2.

What Zhang proved is that there are infinitely many consecutive primes differing by a bounded amount. Zhang’s bound was 70,000,000, but others—James Maynard (Magdalen College Oxford) and the Polymath Project’s many contributors—have since lowered that to 246.

Jointly responsible for the Goldston-Pintz-Yıldırım method for finding small gaps between consecutive primes, Goldston is no stranger to the bounded gap problem. After outlining the GPY method, he indicated to his Carriage House audience some of the psychological and mathematical barriers that perhaps prevented the breakthrough Zhang made from happening sooner.

Some mathematicians may have hesitated to attack the bounded gap problem because of its perceived difficulty.

"A lot of us thought it was so hard because a lot of smart people told us it was hard," Goldston said. "Experts who should have been able to figure it out didn’t figure it out and everyone else figured, 'They’re a lot smarter than we are. Why should we spend five years and fail to do what they could do in two weeks?' Zhang didn’t have that prejudice."

Zhang also, according to Goldston, worked the problem from two angles, eventually putting the pieces together.

Goldston expects mathematicians to incrementally improve on the 246 bound, but doesn’t think the twin prime conjecture will succumb to proof anytime soon. That’s just the nature of primes, slow to give up their secrets.—Katharine Merow